Interval exchange transformations and low-discrepancy
- 13 Downloads
In Masur (Ann Math 115(1):169–200, 1982) and Veech (J Anal Math 33:222–272, 1978), it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the question under which conditions the orbits yield low-discrepancy sequences. The case of \(n=2\) intervals corresponds to circle rotation, where conditions for low-discrepancy are well-known. In this paper, we give corresponding conditions in the case \(n=3\). Furthermore, we construct infinitely many interval exchange transformations with low-discrepancy orbits for \(n \ge 4\). We also show that these examples do not coincide with LS-sequences if \(S \ge 2\).
KeywordsLow-discrepancy Interval exchange transformation Uniform distribution Kronecker sequences LS-sequences
Mathematics Subject Classification11B50 11B83 11K38 37E10
The author thanks the anonymous referees for their useful comments.
- 3.Drmota, M., Tichy, R.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997)Google Scholar
- 5.Kakutani, S.: A problem on equidistribution on the unit interval \([0,1[\). In: Measure Theory (Proceedings of the Conference Oberwolfach, 1975), Lecture Notes in Mathematics, vol. 541 (pp. 369–375). Springer, Berlin (1975)Google Scholar
- 7.Larcher, G.: Discrepancy estimates for sequences: new results and open problems. In: Kritzer, P., Niederreiter, H., Pillichshammer, F., Winterhof, A. (eds.) Uniform Distribution and Quasi-Monte Carlo Methods, Radon Series in Computational and Applied Mathematics, pp. 171–189. DeGruyter, Berlin (2014)Google Scholar
- 15.Weiß, C.: On the Classification of LS-sequences, arXiv:1706.08949 (2017)
- 16.Yoccoz, J.-C.: Continued fraction algorithms for interval exchange maps: an introduction. In: Frontiers in Number Theory, Physics, and Geometry I, pp. 401–435. Springer (2006)Google Scholar